Separation of the Monotone NC Hierarchy

We prove tight lower bounds, of up to n , for the monotone depth of functions in monotone-P. As a result we achieve the separation of the following classes. 1. monotone-NC 6 = monotone-P. 2. For every i 1, monotone-NC i 6 = monotone-NC i+1. 3. More generally: For any integer function D(n), up to n (for some > 0), we give an explicit example of a monotone Boolean function, that can be computed by polynomial size monotone Boolean circuits of depth D(n), but that cannot be computed by any (fan-in 2) monotone Boolean circuits of depth less than ConstD(n) (for some constant Const). 1 from monotone-NC 2 was previously known. Our argument is more general: we deene a new class of communication complexity search problems, referred to below as DART games, and we prove a tight lower bound for the communication complexity of every member of this class. As a result we get lower bounds for the monotone depth of many functions. In particular, we get the following bounds: 1. For st-connectivity, we get a tight lower bound of (log 2 n). That is, we get a new proof for Karchmer-Wigderson's theorem, as an immediate corollary of our general result. 2. For the k-clique function, with k n , we get a tight lower bound of (k log n). This lower bound was previously known for k log n AlBo87]. For larger k, however, only a bound of (k) was previously known.